Given a unital associative ring S and a subring R, we say that S is an ideal
(or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum
is direct. In this note we investigate the ideal structure of an arbitrary
ideal extension of an arbitrary ring R. In particular, we describe the Jacobson
and upper nil radicals of such a ring, in terms of the Jacobson and upper nil
radicals of R, and we determine when such a ring is prime and when it is
semiprime. We also classify all the prime and maximal ideals of an ideal
extension S of R, under certain assumptions on the ideal I.